Logica Universalis 8 (1):25-60 (2014)

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Abstract
Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope
Keywords Non-classical logics  completeness  correspondence  countermodels  proof search  labelled sequent calculi  intuitionistic logic  provability logic  multi-modal logics  geometric rules  Sahlqvist
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DOI 10.1007/s11787-014-0097-1
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References found in this work BETA

Proof Analysis in Modal Logic.Sara Negri - 2005 - Journal of Philosophical Logic 34 (5-6):507-544.
Labelled Deductive Systems: Volume 1.Dov M. Gabbay - 1996 - Oxford, England: Oxford University Press.
Proof Analysis in Intermediate Logics.Roy Dyckhoff & Sara Negri - 2012 - Archive for Mathematical Logic 51 (1-2):71-92.
Labelled Non-Classical Logics.Luca Viganò - 2000 - Boston, MA, USA: Kluwer Academic Publishers.

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Citations of this work BETA

Proof Analysis for Lewis Counterfactuals.Sara Negri & Giorgio Sbardolini - 2016 - Review of Symbolic Logic 9 (1):44-75.
Geometrisation of First-Order Logic.Roy Dyckhoff & Sara Negri - 2015 - Bulletin of Symbolic Logic 21 (2):123-163.

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